A new method of DC power supply modelling for rapid transit railway system simulation Z.Y. Shao\ W.S. Chan", J. Allan* & B. Mellitt" Iz'rm'W, ^ The University of Birmingham, UK Introduction The Multi-Train Simulator (MTS) originally developed by the University of Birmingham more than 20 years ago, with an investment in excess of 30 person years, was purchased by London Underground Limited (LUL) in 1989 [1, 2, 3]. The MTS is capable of modelling all the major electrical and mechanical subsystems on rapid transit railways including power supply, rolling stock, signalling and control systems and the interactions between these systems. Therefore, the MTS has been extensively used as a main design tool on mega engineering projects such as the Central Line modernisation project and the Jubilee Line Extension project. At the same time, a multi-stage development programme has been initiated to further develop the MTS to become an integrated railway simulation package encompassing all railway aspects. One of the important areas of the development programme was to review the existing power network solving technique and its suitability for multi-track network with interconnection between all tracks. This was with a view to develop a more versatile power network solution technique which will result in savings in computation time and memory storage. This work can form the foundation for future plans of providing real-time monitoring and control facility using what-if predictive simulation adopting parallel computing techniques. Typical LUL applications The MTS has been used in the following power system studies by LUL:
552 Railway Design and Management a) Northern Line substation renewal - quantitative assessment in order to assist the Northern Line project team in defining a system specification for substation design. b) Jubilee Line Extension (JLE) harmonics study - ac supply harmonic analysis. c) Jubilee Line power tender assessment - used as a design tool to verify power design calculations submitted by the tenderers. d) Central Line renewal project - used to assess dc voltage regulation due to the possible introduction of voltage controlled rectifier substations. e) Traction voltage optimisation study - used to study the relationship between energy consumption and traction voltage. Based on the amount of application work already carried out, it is important to ensure that an efficient power network solution is used. Areas considered Key areas of power network solution processes considered include: a) Circuit analysis technique - nodal or mesh. b) Optimisation of electrical network topology numbering technique. c) Matrix solver. d) Iteration. As a result of the review in the above areas a new general model for DC power feeding systems with interconnection between tracks has been established. The flexible topological approach adopted based on the concept of Current Rail Section (CRS) enables the network to be established node by node, each node being either a train, a feeder or a junction. Each current rail section refers to a unidirectional feeding section from a substation. The equivalent power conductor rail resistance between adjacent nodes can be readily derived. A node numbering method has also been established to assemble the network into a symmetrical sparse matrix. A fast sparse matrix decomposition and solving technique has also been derived. A new integration technique on checking the non-receptivity of substations and other system constraints together with the adoption of the relaxation method ensures that system stability and required accuracy are achieved. Review of existing MTS methods To fully recognise the effectiveness of the new method it will be useful to understand the existing method of solving the power network. For a two-road network shown in figure 1, substations and trains are represented by their equivalent voltage sources and resistances. This has resulted in two parallel ladder networks connected at the substations and track coupling points. As the
Railway Design and Management 553 interconnections are clearly defined and all the current loops are easily identified, a mesh analysis technique has been adopted. An equivalent symmetrical matrix can be derived due to the inherent feature of any ladder network. To assemble all the loop currents into its matrix form, a systematic way of numbering the current loops which reduces the band width of the matrix has been derived. Basically, the numbering of loops is carried out by alternating between up and down roads with a view to minimise the separation between two trains on up and down roads connecting to the same substation. A detailed description of this numbering method can be found in reference 4. A narrow band symmetrical matrix will result from this method with the band width solely dependent on the difference between the number of up and down road trains situated between adjacent substations. The Cholesky matrix decomposition method is then used to solve the network followed by forward and backward substitution. As with the Cholesky method, only the elements of the widest band width and all other elements in each row (see reference 4) are required to be stored for matrix manipulation, the size of the resultant matrix and hence the speed of computation is largely dependent on the maximum band width of the network considered. Disadvantages of the existing method for Multi-Track Application The main problem with the existing method is that the current loop numbering scheme which is efficient for a simple two road network can no longer be applied effectively for a multi-track network with interconnection between all tracks. This is due to the separation between trains and substations now being much wider which will result in a much wider band matrix. In most cases, the band width of the matrix is getting close to the dimension of the full matrix. Hence, the Cholesky decomposition becomes inefficient as it is only suitable for a narrow band matrix. The other disadvantage of the existing method for multi-track application is the use of the mesh analysis technique. As branches and loops from the main section of the line need to be modelled (see figure 2), it is more difficult to define all the independent loop currents. If more branches and junctions have to be considered, the advantages of using nodal analysis become more apparent, not only the nodes of the network are more easily identified but also the number of node equations becomes less than that of the mesh equations. New method of assembly power network With reference to figure 2, each node represents a connection to a substation, a train or a junction. The current entering and leaving each node can be described by using the KirchhofFs current law. The way of numbering the nodes is the key in optimising the equivalent matrix. It was found that the minimum number of elements required to be stored in the decomposed matrix can be obtained by
554 Railway Design and Management starting the nodal equations from the branch with the least number of trains between two substations followed by the branch with next least number of trains and so on. The node numbering scheme is clearly shown in figure 2. By arranging the resultant nodal equations in matrix form, an equivalent resistance matrix representation can be established as shown in figure 3. As the matrix is symmetrical about the diagonal elements, only the bottom left half needs to be stored. Furthermore, with the customised sparse matrix using the L-U factorisation technique developed in-house, only the non-zero elements below the diagonal elements (the asterisks shown in figure 3) will be decomposed and stored into a one dimensional matrix array as highlighted in figure 3. With this node numbering scheme, the number of elements to be stored is always equal to the maximum number of trains and substations which is useful for checking if the optimised storage has been achieved. As only the non-zero elements are required to be decomposed to a one dimensional matrix, this has resulted in further reduction in memory storage and hence computation time compared with the existing Cholesky method. This technique has been applied to a complex network such as the Northern Line shown in Figure 4. Necessity for iterative process It can be shown that under normal conditions, the solution in each update obtained by solving the power network once is sufficiently accurate after satisfying other non-linear constraints of the system, eg. receptivity of the substation and over voltage and under voltage trains. This has been the case for the existing MTS. However, with the requirements of interfacing the dc network with the 22/1 IkV ac network, it is necessary to consider an outer iteration process which has been incorporated in the network solution to achieve the required accuracy of the solution. The criteria used for convergence test is the Voltage mismatch' which is the same as that used commonly in load flow analysis. The accuracy test is performed by comparing the nodal voltages as successive integration until the required accuracy is achieved. System stability When regenerative braking is considered in the simulation, limit cycle oscillation might occur in the system. This is because there are insufficient motoring trains to absorb the energy regenerated. Since non-inverting traction substations are used, the traction voltage could be raised above the maximum limited voltage level, should there be no proper functional over-voltage controller equipment on board the regenerating train. For the purpose of simulation, an ideal over-voltage controller is assumed to be installed on each train. It assumes that each regenerative braking train is capable
Railway Design and Management 555 of dissipating part or all of its kinetic energy for a desirable braking profile should the traction voltage reach the specified voltage limit. The traction voltage at the regenerative braking train position can stay constant at the maximum level as long as the regenerative braking train has enough energy to return back to the network. In dealing with non-receptive substations, the existing method models the substation off state by an equivalent large resistance of 1000Q in series with the substation no load voltage. LUL utilise a four-rail system with positive and negative power rail. Each pole of the system is earthed via one bleed resistor at each substation which has resulted in two resistors connected in series between the positive and negative traction conductors with the middle point earthed. This feature enables a better representation of a nonreceptive substation model for simulation. If the substation is simply modelled as a huge resistance, the whole system turns too stiff if an iteration process is used. The simulation will be prolonged due to difficulty in the iteration process. This problem can be solved by a sectional relaxation based method together with the use of thefiniteleakage resistance as described above. Assuming 300(1 is the value of bleeding resistance referred to each substation, at the maximum voltage limit (for example, at 790V) the bleeding current will be 2.6 Amperes. This figure is used as tolerable negative current for each substation. So the internal resistance can be simply specified as (790-670)72.6 = 46Q with the substation no-load voltage staying at 670V when the substation becomes non-receptive Since the regenerative braking trains are clamped to a maximum voltage limit, the bleeding current can be maintained at this small value. Example of Simulation An example of the output graphic facilities on the section north of Camden Town of the Northern Line is shown in figure 5. The network consists of three DC sections and fourteen substations with maximum 76 trains on the sections. The simulation was carried out on a SUN IPX-Workstation and the computation time for an hour simulation was twelve minutes which is approximately one-sixth of the time achieved by the previous MTS. Conclusions a) A new method of dc power supply modelling and network solution has been presented for older metro lines with complex junctions and loops, such as LUL. b) The key features of the new model can be summarised as: nodal based method optimised node numbering scheme
556 Railway Design and Management (iii) (iv) in-house customised sparse matrix solver new iteration technique c) It was found that the new method is more efficient for the type of application considered. One sixth of the original computation time was achieved for the case on north of Camden Town of the Northern Line considered. d) It is planned to extend this method to include the use of parallel computing techniques so that real time simulation on the whole LUL network can be achieved. References Mellitt, B, Goodman, C.J. & Arthurton, RIM 'Simulator for studying Operational and Power Supply Conditions in Rapid Transit Railways', Proc. IEE, Vol. 125, 4, pp 288-303, 1978. Allan, J, Chan, W.S., Mellitt, B, Anderson, P. & Chaing, J.P. 'Developments in Multi-Train Simulation by London Underground Limited and Hong Kong Mass Transit Railway Corporation', International Conference on Computer Aided Design and Manufacture and Operations in the Railway and other advanced Mass Transit Systems, CompRail '92, August 1992. Allan, J., Digby, G, Chan, W.S. & McCormick, H.J. 'Cost Conscious Design of Power Supply Equipment using a Simulator', IEE Colloquium on Cost Effective Industrial Simulation, June 1990. Rambukwella, N.B., Mellitt, B, Goodman, C.J. & Mouneimne, Z.S. Traction Equipment Modelling and the Power Network Solution for DC supplied Traction Power Systems studies', IEE International Conference on Electric Railway Systems for a new century', pp 218-224, 1987. Figure 1: Loop numbering scheme for two road network
Railway Design and Management 557 8 <D rm Figure 2: An example of the new node assigning scheme 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 * 3 4 5 6 7 9 10 11 12 13 14 15 16 Figure 3: Equivalent sparse matrix of power network shown in Figure 2
558 Railway Design and Management SUBSTATION LAYOUT FOR NORTHERN LINE DC 5 DC 4 Figure 4: Power network for existing Northern Line LONDON UNDERGROUND LTD Figure 5: Example of MTS output